[Mesa-dev] [PATCH 5/8] glsl: Rewrite atan2 implementation to fix accuracy and handling of zero/infinity.

[Francisco Jerez]

This addresses several issues of the current atan2 implementation:

 - Negative zero (and negative denorms which end up getting flushed to
   zero) isn't handled correctly by the current implementation.  The
   reason is that it does 'y >= 0' and 'x < 0' comparisons to decide
   on which side of the branch cut the argument is, which causes us to
   return incorrect results (off by up to 2π) for very small negative
   values.

 - There is a serious precision problem for x values of large enough
   magnitude introduced by the floating point division operation being
   implemented as a mul+rcp sequence.  This can lead to the quotient
   getting flushed to zero in some cases introducing an error of over
   8e6 ULP in the result -- Or in the most catastrophic case will
   cause us to return NaN instead of the correct value ±π/2 for y=±∞
   and x very large.  We can fix this easily by scaling down both
   arguments when the absolute value of the denominator goes above
   certain threshold.  The error of this atan2 implementation remains
   below 25 ULP in most of its domain except for a neighborhood of y=0
   where it reaches a maximum error of about 180 ULP.

 - It emits a bunch of instructions including no less than three
   if-else branches per scalar component that don't seem to get
   optimized out later on.  This implementation uses about 13% less
   instructions on Intel SKL hardware and doesn't emit any control
   flow instructions.
---
 src/compiler/glsl/builtin_functions.cpp | 82 ++++++++++++++++++---------------
 1 file changed, 46 insertions(+), 36 deletions(-)

diff --git a/src/compiler/glsl/builtin_functions.cpp b/src/compiler/glsl/builtin_functions.cpp
index 4a6c5af..fd59381 100644
--- a/src/compiler/glsl/builtin_functions.cpp
+++ b/src/compiler/glsl/builtin_functions.cpp
@@ -3560,44 +3560,54 @@ builtin_builder::_acos(const glsl_type *type)
 ir_function_signature *
 builtin_builder::_atan2(const glsl_type *type)
 {
-   ir_variable *vec_y = in_var(type, "vec_y");
-   ir_variable *vec_x = in_var(type, "vec_x");
-   MAKE_SIG(type, always_available, 2, vec_y, vec_x);
-
-   ir_variable *vec_result = body.make_temp(type, "vec_result");
-   ir_variable *r = body.make_temp(glsl_type::float_type, "r");
-   for (int i = 0; i < type->vector_elements; i++) {
-      ir_variable *y = body.make_temp(glsl_type::float_type, "y");
-      ir_variable *x = body.make_temp(glsl_type::float_type, "x");
-      body.emit(assign(y, swizzle(vec_y, i, 1)));
-      body.emit(assign(x, swizzle(vec_x, i, 1)));
-
-      /* If |x| >= 1.0e-8 * |y|: */
-      ir_if *outer_if =
-         new(mem_ctx) ir_if(greater(abs(x), mul(imm(1.0e-8f), abs(y))));
-
-      ir_factory outer_then(&outer_if->then_instructions, mem_ctx);
-
-      /* Then...call atan(y/x) */
-      do_atan(outer_then, glsl_type::float_type, r, div(y, x));
-
-      /*     ...and fix it up: */
-      ir_if *inner_if = new(mem_ctx) ir_if(less(x, imm(0.0f)));
-      inner_if->then_instructions.push_tail(
-         if_tree(gequal(y, imm(0.0f)),
-                 assign(r, add(r, imm(M_PIf))),
-                 assign(r, sub(r, imm(M_PIf)))));
-      outer_then.emit(inner_if);
-
-      /* Else... */
-      outer_if->else_instructions.push_tail(
-         assign(r, mul(sign(y), imm(M_PI_2f))));
+   const unsigned n = type->vector_elements;
+   ir_variable *y = in_var(type, "y");
+   ir_variable *x = in_var(type, "x");
+   MAKE_SIG(type, always_available, 2, y, x);
 
-      body.emit(outer_if);
+   /* If we're on the left half-plane rotate the coordinates π/2 clock-wise
+    * for the y=0 discontinuity to end up aligned with the vertical
+    * discontinuity of atan(s/t) along t=0.
+    */
+   ir_variable *flip = body.make_temp(glsl_type::bvec(n), "flip");
+   body.emit(assign(flip, less(x, imm(0.0f, n))));
+   ir_variable *s = body.make_temp(type, "s");
+   body.emit(assign(s, csel(flip, abs(x), y)));
+   ir_variable *t = body.make_temp(type, "t");
+   body.emit(assign(t, csel(flip, y, abs(x))));
 
-      body.emit(assign(vec_result, r, 1 << i));
-   }
-   body.emit(ret(vec_result));
+   /* If the magnitude of the denominator exceeds some huge value, scale down
+    * the arguments in order to prevent the reciprocal operation from flushing
+    * its result to zero, which would cause precision problems, and for s
+    * infinite would cause us to return a NaN instead of the correct finite
+    * value.
+    */
+   ir_constant *huge = imm(1e37f, n);
+   ir_variable *scale = body.make_temp(type, "scale");
+   body.emit(assign(scale, csel(gequal(abs(t), huge),
+                                imm(0.0625f, n), imm(1.0f, n))));
+   ir_variable *rcp_scaled_t = body.make_temp(type, "rcp_scaled_t");
+   body.emit(assign(rcp_scaled_t, rcp(mul(t, scale))));
+   ir_expression *s_over_t = mul(mul(s, scale), rcp_scaled_t);
+
+   /* Calculate the arctangent and fix up the result if we had flipped the
+    * coordinate system.
+    */
+   ir_variable *arc = body.make_temp(type, "arc");
+   do_atan(body, type, arc, abs(s_over_t));
+   body.emit(assign(arc, add(arc, mul(b2f(flip), imm(M_PI_2f)))));
+
+   /* Rather convoluted calculation of the sign of the result.  When x < 0 we
+    * cannot use fsign because we need to be able to distinguish between
+    * negative and positive zero.  Unfortunately we cannot use bitwise
+    * arithmetic tricks either because of back-ends without integer support.
+    * When x >= 0 rcp_scaled_t will always be non-negative so this won't be
+    * able to distinguish between negative and positive zero, but we don't
+    * care because atan2 is continuous along the whole positive y = 0
+    * half-line, so it won't affect the result.
+    */
+   body.emit(ret(csel(less(min2(y, rcp_scaled_t), imm(0.0f, n)),
+                      neg(arc), arc)));
 
    return sig;
 }
-- 
2.10.2